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====== Monogamy 1 Gender ======

Analysis and numerics for case of monogamy.  Idealized to a single gender for convenience but (as was found for [[..:2009:Estrous 2 Genders | Estrous]]) we expect the results to generalize to two genders.  We find here that monogamy simplifies population dynamics in sexual populations to the logistic rate equation (for large populations, with a weak Allee effect for small populations).  

===== Processes =====

Let \(S\) represent a single individual (singleton) and \(P\) indicate a bonded pair.

|  \( S + S \xrightarrow{\phi} P \)  | pair bonding  |
|  \( P \xrightarrow{\chi} 2 S \)  | separation  |
|  \( P \xrightarrow{\beta} S + P \)  | birth  |
|  \( S + S \xrightarrow{\delta} S \)  | competition between singletons  |
|  \( S + P \xrightarrow{2 \delta} P \)  | competition between singleton and pair removes singleton  |
|  \( P + S \xrightarrow{2 \delta} 2 S \)  | competition between pair and singleton removes pair member  |
|  \( P + P \xrightarrow{4 \delta} S + P \)  | competition between pairs  |

The first two are strictly behavioural processes (transitions between pair-bonded and singleton states) while the remainder are ecological (affecting the population density).

==== Competition processes ====

The competition processes require some explanation.  Let's assume pair members don't compete with each other but pairs do compete with each other and with singletons.  If we assume members of bonded pairs each have the same risk from competition as singletons (pairing doesn't offer protection or increase risk) then there are multiple "ways" for competition to affect pairs.  For example, between two pairs there are four possible competitive interactions that could occur (each member of the focal pair can be out-competed by each member of the other) so the rate is increased by a factor of four.

===== Full Dynamics =====

\[
  \diff{s}{t} = - 2 \phi s^2 + 2 \chi p + \beta p -\delta s^2 - 2 \delta s p + 2 \delta s p + 4 \delta p^2 
\]
\[
  \diff{p}{t} = + \phi s^2 - \chi p - 2 \delta s p - 4 \delta p^2 
\]
\[
  \diff{n}{t} = + \beta p - \delta n^2 
\]

where \(n=s+2 p\).
 
===== Slow Timescale =====

==== Units ====

Let's assume that ecological processes are much slower than behavioural ones.  In particular, let's assume

\[
  \beta, \delta \abs{K} \ll \phi \abs{K}, \chi 
\]

where \( \abs{K} \) is a population density constant, ie. it has the same units as \(n\), \( [K] = [n] \) (where \( [\cdot] \) means "units of").  We choose \( \abs{K} \) so it is a characteristic population size for the dynamics.  That way we can compare the rate constants, because \( K \) matches the units: 

\[
  [\beta ] = [\delta K] = [\phi K] = [\chi]. 
\]

==== Separation of Timescales ====

We can now specify what we mean when we say "ecological processes are much slower than behavioural ones".  Let us define \( \hat{\beta} \) and \( \hat{\delta} \) so that

\[
  \beta \equiv \epsilon \hat{\beta}, 
\]
\[
  \delta \equiv \epsilon \hat{\delta} / \abs{K} 
\]

where \(\epsilon \ll 1 \) is a dimensionless constant.  Since \(\epsilon \ll 1\) the ecological processes (involving \(\beta\) and \(\delta\)) occur much less frequently than the behavioural transitions (with \(\phi\) and \(\chi\)).

Now we can look at the dynamics under our original (fast) time variable \(t\) or with a new, slow, time variable, \( \hat{t} \equiv \epsilon t \).  In this slower timescale we find the dynamics become

\[
  \epsilon s' \equiv \epsilon \diff{s}{\hat{t}} = -2 \phi s^2 + 2 \chi p + \epsilon \hat{\beta} p - \epsilon \frac{\hat{\delta}}{\abs{K}} (s^2 + 4 p^2) 
\]
\[
  \epsilon p' \equiv \epsilon \diff{p}{\hat{t}} = + \phi s^2 - \chi p - 2 \epsilon \frac{\hat{\delta}}{\abs{K}} (s + 2 p) p 
\]
\[
  n'          \equiv \diff{n}{\hat{t}} = \hat{\beta} p - \frac{\hat{\delta}}{\abs{K}} n^2. 
\]

where the prime, \( \cdot' \), denotes the slow derivative, \( \diff{\cdot}{\hat{t}} \).

Note that \(\epsilon\) cancels out in the dynamical equation for \(n\) which means that \(n\) varies at a measurable rate on the slow timescale--\(n\) is a ''slow'' variable.

===== Quasi-Steady-State Assumption =====

So far our analysis has been exact.  Now let's assume that the other, ''fast'' variables equilibrate quickly so that \( \epsilon s' \approx 0 \approx \epsilon p' \), the quasi-steady-state assumption, QSSA.  Then we can compute the quasi-steady-state densities \(s\) and \(p\) at any time.  Written in terms of the fast parameters (\(\beta\) and \(\delta\)) they are

<div right round 50% tip>The derivation can be found in ''monogamy1g20091106.nb''.</div>

\[
  s \approx \frac{-\beta +4 n \delta -2 \chi +\sqrt{(\beta -4 n \delta +2 \chi )^2+16 n (\delta +\phi ) (\beta -2 n \delta +2 \chi )}}{8 (\delta +\phi )} 
\]
\[
  p \approx \frac{2 n \delta +4 n \phi +\chi -\sqrt{ (2 n \delta +\chi ) (2 n \delta +8 n \phi +\chi )}}{8 \phi }.
\]

Fortunately, this gives us a way to check the validity of our assumption: we originally required \(s+2p=n\) so this should still be largely correct.  Approximating as a Taylor series in \(n\) gives the first-order correction:

\[
  s+2p \approx n \left[ 1 + \frac{2 \beta  \phi -2 \delta  \chi }{\beta  \chi +2 \chi ^2} n + \ldots \right] \equiv n \left[ 1 \pm n/n_\epsilon + \ldots \right]. 
\]

This tells us the QSSA may only be valid when

\[
  n \ll n_\epsilon = \frac{\beta  \chi +2 \chi ^2}{\abs{2 \beta  \phi -2 \delta  \chi }}. 
\]

==== Alternate QSSA Condition ====

<div right round 50% tip>What about \( \beta \ll \phi \abs{K} \)?  That puts a lower bound on \( \abs{K} \): \( \abs{K}\gg \beta/\phi \). </div>

For separation of timescales we assumed \( \delta \abs{K} \ll \chi \) (among other things).  That puts a condition on \(\abs{K}\) for QSSA to be satisfied:

\[
  \abs{K} \ll Q \equiv \frac{\chi}{\delta}. 
\]
 
===== QSSA Birthrate =====

When the QSSA holds we can approximate the population dynamics on the fast, behavioural timescale by

\[
  \diff{n}{t} \approx B(n) n - \delta n^2 
\]

where the per-capita birthrate, \(B(n)\), is
\[
  B(n) \equiv \frac{\beta}{8 \phi n } \left( 2 n (\delta +2 \phi )+\chi - \sqrt{ (2 n \delta +\chi ) (2 n \delta +8 n \phi +\chi )} \right). 
\]

\(B(n)\) is a monotonically increasing function that climbs from \(B(0)=0\) to 
\[
  r \equiv B(\infty) = \frac{\beta}{4 \phi }  \left(\delta +2 \phi -\sqrt{\delta  (\delta +4 \phi )}\right). 
\]

<div right round box>
{{monogamy1gSpecificBirthrate20091210.jpg}}

Figure 1: Per-capita birthrate, \(B(n)\) for population dynamics of \(n\) given QSSA assumption.  The function looks qualitatively similar to the estrous birthrate curve.  (Notice that \(\beta > 2 r\) always.)
</div>

<div right round 50% tip>The derivation can be found in ''monogamy1gSpecificBirthrate20091210.nb''.</div>

We define the Allee domain to be \(n<n_\mathrm{Allee}\) where \(B(n_\mathrm{Allee})=r/2\) which gives a solution

\[
  n_\mathrm{Allee} = \frac{\delta +2 \phi -3 \sqrt{\delta  (\delta +4 \phi )} }{2 \phi ^2 - 16 \delta \phi - 4 \delta ^2 } \chi . 
\]

==== Logistic domain ====

If \(n_\mathrm{Allee} \ll n \ll n_\epsilon\) then the dynamics lie within the ''logistic domain'' where the per-capita birthrate is roughly constant, \(B(n)\approx r\), and the population dynamics follow

\[
  \diff{n}{t} \approx r n - \delta n^2, 
\]

the logistic rate equation.

==== Allee domain ====

On the other hand, if the population is small, \( n \ll n_\mathrm{Allee} \), then the birthrate is suppressed, the Allee effect.  In this domain the per-capita birthrate is roughly

\[
  B(n)\approx \frac{r \beta^2}{\beta^2 - 2 r^2} \frac{n}{n_\mathrm{Allee}} 
\] 

and the population follows

\[
  \diff{n}{t} \approx \left( \frac{r \beta^2}{ \left( \beta^2 - 2 r^2 \right) n_\mathrm{Allee}} - \delta \right) n^2. 
\]

Notice the population is unstable if \( r \beta^2 < \delta n_\mathrm{Allee} \left( \beta^2 - 2 r^2 \right) \) or, equivalently, \( \beta \phi < \delta \chi \).

===== Carrying capacity =====

Regardless of whether QSSA holds we can find the non-trivial equilibrium population density. Since this represents the ''characteristic'' population size we can choose this to represent our parameter \(K\) from above:

\[
  K \equiv \frac{1}{4 (\beta - 2 r)^2 \phi }\left(4 r^2 \beta +\beta ^3 - (\beta-2 r) \sqrt{\beta  \left(\beta  (2 r+\beta )^2+8 n_\mathrm{Allee} \left(\beta ^2 - 2 r^2\right) \phi \right)} \right). 
\]

I haven't done the Jacobian stability analysis but it looks pretty clear that \(n=K\) is only stable when \(K>0\), otherwise only \(n=0\) is stable.  Then if \(K\leq 0\) the population is unstable and will inevitably go extinct.  So we only expect \(K>0\) to represent naturally-occurring populations (unless the parameters have changed recently to destabilize the population).

If \(K>0\) then the ''weak'' Allee effect applies in the Allee domain so the population is globally stable (regardless of how small it gets).  On the other hand, if \(K<0\) then the system is unstable for all \(n\).

One final point about the carrying capacity.  For comparison with the logistic equation, in terms of \(r\), \(K\), and \(n_\mathrm{Allee}\) the QSSA dynamics can be written

<div right round 50% tip>Derivation in ''monogamy1gPrettifyRate20091218.nb'' </div>

\[
  \diff{n}{t} = \frac{r}{4} \left(2 n - n_\mathrm{Allee} - \frac{n^2 \left(\beta^2 (2 K - n_\mathrm{Allee}) + 8 (K+n_\mathrm{Allee}) r^2\right)}{\beta^2 K^2} \right. 
\]
\[
  \left. + \frac{8 (n + n_\mathrm{Allee}) r^2}{\beta ^2} + \frac{n^2 f(K)  -K^2 f(n)}{K^2}\right) 
\]
where
\[
  f(x) \equiv \sqrt{ (n_\mathrm{Allee} - 2 x)^2 + \frac{64 r^4 (n_\mathrm{Allee} + x)^2}{\beta ^4} - \frac{16 r^2 \left(n_\mathrm{Allee}^2 - n_\mathrm{Allee} x + 2 x^2\right)}{\beta^2}}. 
\]

In the limit \(n_\mathrm{Allee} \ll n, K\) this reduces to the familiar logistic equation,
\[
  \lim_{n_\mathrm{Allee}\rightarrow 0} \diff{n}{t} = r n \left( 1 - \frac{n}{K} \right). 
\]

===== Small epsilon =====

The above analysis is problematic because it doesn't reveal important regions of parameter space that should be checked for accuracy of the QSSA.  The equations are too complex.  But we can make progress if we accept that the parameter \(\epsilon\) is small so we can do a Taylor expansion around \(\epsilon=0\) and just keep the lower order terms.

<div right round 50% tip>See monogamy1gSmallEpsilon20091227.nb for derivation. </div>

Then we find


\[
  s \approx \frac{-\chi + \sqrt{ \chi(8 n \phi + \chi) } }{4 \phi} 
\]

\[
  p \approx \frac{n}{2} + \frac{\chi - \sqrt{ \chi(8 n \phi + \chi) } }{8 \phi} 
\]

\[
  n_\epsilon \approx \frac{\chi^2}{\abs{\beta \phi - \delta \chi}} 
\]

\[
  B(n) \approx \frac{\beta}{2} + \frac{\beta}{8 n \phi} \left( \chi - \sqrt{ \chi(8 n \phi + \chi) } \right) 
\]

\[
  r \approx \frac{\beta}{2} 
\]

\[
  n_\mathrm{Allee} \approx \frac{\chi}{\phi} 
\]

\[
  K \approx \frac{\beta}{2 \delta} \left( 1 - \sqrt{\frac{\delta \chi}{\beta \phi}} \right). 
\]

From all of that the dynamics of \(n\) reduce to 

\[
  \diff{n}{t} \approx r n \left( 1 - \frac{n}{K} \right) + \frac{r}{4 n_\mathrm{Allee} K^2} \left( n^2 f(K) - K^2 f(n) \right) 
\]

where

\[
  f(x) \approx \sqrt{1 + 8 x n_\mathrm{Allee}} - 1. 
\]
 
===== Parameter space =====

The possible dynamics (of the small-\(\epsilon\) approximation) can be expressed more concisely by reducing the parameter space to two dimensions: 

\[
  x=\delta/\phi 
\]
\[
  y=\beta/\chi. 
\]

We will find there are just three important regions in this parameter space, depending on the relationships between the constants \(n_\epsilon, K\), and \(n_\mathrm{Allee}\).  In terms of \(x\) and \(y\) these constants can be written (for small \(\epsilon\) as

\[
  n_\epsilon = \frac{\chi}{\phi} \frac{1}{\abs{y-x}} 
\]
\[
  K = \frac{\chi}{\phi} \frac{1}{2x} ( y - \sqrt{xy} ) 
\]
\[
  n_\mathrm{Allee} = \frac{\chi}{\phi}. 
\]

==== Stability ====

If \(K\leq 0\) then we expect the population to be unstable and to crash (under both the full dynamics and the QSSA approximation).  In terms of \((x,y)\) this is


|  Unstable :  | \( y\leq x. \)  |


==== QSSA limit ====

If \( K > 0\)  (\(y>x\)) then \(K\) represents a characteristic population density (actually, the equilibrium) so it is reasonable to expect the actual density to be on the same order, \( n \sim K \).  Then we can express the QSSA condition as \( K \ll n_\epsilon \).  This condition implies

\[
  \frac{1}{2x} ( y - \sqrt{xy} ) \ll \frac{1}{\abs{y-x}} 
\]

or, approximately


|  QSSA :  | \( y \ll \sqrt{2x} + \left( \frac{x}{2} \right)^{3/4} + \frac{3x}{4}. \)  |


==== Logistic vs Allee domain ====

If \(K>0\) then the dynamics will carry \(n\) towards \(K\).  If \(K<n_\mathrm{Allee}\) then the population will always be in the Allee domain with a depressed population growth rate.  This condition is simply


|  Allee:  | \( y < 4 x. \)  |


==== Summary ====

<div right round 50% tip>It seems natural to me but I don't know how to justify \(x,y \lesssim 1\).  Does it follow from \(\epsilon\ll 1\)? </div>

<div right round box>
{{:research:draft:2010:monogamy1genderparameterspace.png}}

Figure 2: Parameter space can be divided into four regions characterizing the dynamics.  The red dots indicates points where the dynamics are computed numerically (see below).
</div>

In conclusion, there are four regions of interest in the \((x,y)\)-plane:

|  stable non-QSSA :   | \( K \gtrsim n_\epsilon > 0 \)  |
|  stable logistic QSSA :   | \( n_\epsilon \gtrsim K \gtrsim n_\mathrm{Allee} > 0 \)  |
|  stable Allee QSSA :   | \( n_\epsilon, n_\mathrm{Allee} \gtrsim K > 0 \)  |
|  unstable :   | \( 0 > K. \)  |

===== Numerics =====

To test our analytic results we can construct a numeric simulation.  The above analysis suggests the regions we should check numerically (marked in red on Figure 2).  The results, shown below, suggest the QSSA is a more robust approximation than would be expected from the condition \(n\ll n_\epsilon\).  

|   Region     |   \( x = 0.05 \)   |   \( x = 0.2 \)   |   \( x = 0.8 \)   |
|   Non-QSSA \\ \( K > n_\epsilon, n_{Allee} > 0 \)   |  {{monogamy1g20100108_stablenon_x0.05y0.8.png?200}}   |   {{monogamy1g20100108_stablenon_x0.2y1.5.png?200}}   |      |
|   Logistic \\ \( n_\epsilon > K > n_{Allee} > 0 \)   |   {{monogamy1g20100108_logistic_x0.05y0.3.png?200}}   |   {{monogamy1g20100108_logistic_x0.2y0.9.png?200}}   |      |
|   Allee \\ \( n_\epsilon > n_{Allee} > K > 0 \)   |   {{monogamy1g20100108_allee_x0.05y0.1.png?200}}   |   {{monogamy1g20100108_allee_x0.2y0.5.png?200}}   |   {{monogamy1g20100108_allee_x0.8y1.5.png?200}}   |
|   Unstable \\ \( n_\epsilon, n_{Allee} > 0 > K \)   |  {{monogamy1g20100108_unstable_x0.05y0.025.png?200}}   |   {{monogamy1g20100108_unstable_x0.2y0.1.png?200}}   |   {{monogamy1g20100108_unstable_x0.8y0.4.png?200}}   |

==== Small epsilon approximation ====

For completeness I have also plotted the dynamics for the small \(\epsilon\) approximation.  We should expect the approximation to fit well when \(x\ll 1\) and/or \(y\ll 1\) because \(x,y \propto \epsilon\).  As expected, the numerics show strong correspondence when either \(x\) or \(y\) is small.  But the general QSSA approximation is always a better description of the full dynamics.  For example, in many cases the small \(\epsilon\) carrying capacity \(K\) varies significantly from the actual carrying capacity.
